Question

# If - 5,k, - 1 are in AP then the values of k is equal to:$A. - 5 \\ B. - 3 \\ C. - 1 \\ D.{\text{ }}3 \\ E.{\text{ 5}} \\$

Hint- This question is solved by using the formula for ${n^{th}}$ term when a series is in AP.

Now,
Given that $- 5,K, - 1$ are in AP.
And we have to find the value of $k$ .
Now we know the formula to find the ${n^{th}}$ of an AP
${a_n} = {a_1} + \left( {n - 1} \right)d$
Here ${a_n}$ is the ${n^{th}}$ term,
${a_1}$ is the first term,
$d$ is the common difference and
$n$ is the number of terms which are to be found.
Now,
${a_3} = {a_1} + \left( {n - 1} \right)d$
Here, ${a_3} = - 1$ , ${a_1} = - 5$, $n = 3$ and
$d = k - \left( { - 5} \right) \\ = k + 5 \\$
Putting the value of these we get,
$- 1 = - 5 + \left( {3 - 1} \right)\left( {k + 5} \right) \\ {\text{or }} - 1 = - 5 + 2\left( {k + 5} \right) \\ {\text{or }} - 1 + 5 = 2k + 10 \\ {\text{or }}4 = 2k + 10 \\ {\text{or }}4 - 10 = 2k \\ {\text{or }} - 6 = 2k \\ {\text{or }}k = - 3 \\$
Thus, the correct option is $\left( B \right)$.

Note- Whenever we face such types of questions the key concept is that we should know the formulas when a series is in AP. Like we did in this question here, we apply the formula for ${n^{th}}$ term and we find the solution.